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Indexed sets
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Indexed sets
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Summation notation“Recall” that we can write a long sum of a bunch of numbers usingsummation notation.
a1 + a2 + · · · + an =nÿ
i=1ai
We can even write infinite sums:
1 + 12 + 1
4 + · · · + 12i + · · · =
Œÿ
i=1
12i
n number G taztast tan
O4 7.1 8 13 11
i takes orneryvalue in IN
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Indexed setsSuppose we have a bunch of sets A1, A2, . . . , An. Then we canwrite:
A1 fi A2 fi · · · fi An =n€
i=1Ai
and
A1 fl A2 fl · · · fl An =n‹
i=1Ai
all sets
I
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Indexed setsIf A1, A2, . . . , An are all sets, then
n€
i=1Ai = {x : x belongs to at least one set Ai}
I A1 = {1, 4, 10, 12}I A2 = {5, 12, 15}I A3 = {1, 4, 15, 35}
What ist3
i=1 Ai?
OAi A VANAY
xc A.IE Is
4 1,440,12 515,35
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Indexed setsn‹
i=1Ai = {x : x belongs to every set Ai}
I A1 = {1, 4, 10, 12}I A2 = {5, 12, 15}I A3 = {1, 4, 15, 35}
What isu3
i=1 Ai?
IT F Ai _A nAzhA31 Ict 12 SAA
4 Az ISKA
NEEL'T ff 354A
f
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Indexed setsOne can also take the union and intersection of infinitely many sets.tŒ
i=1 Ai anduŒ
i=1 Ai .
Example. For each i œ N, let
Ai = {≠i , 0, i}
What istŒ
i=1 Ai anduŒ
i=1 Ai .?
A a boy EZ
A z 2,0 2 C 2As 3,0 3 EZ
atleast
eAi x x belongs bone of Ai
Any integer belongsto some Ai
Take an integer n Ten n c An f h QuSo every integer belongs
to Ai
so2Ai _f x x belongs toevery Ai Onlyzerohasth proserly A Il Tt
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Index setsInstead of numbering the sets, one can label them with elements ofany set I called an index set.t
iœI Ai is the set of elements that belong to at least one of the setsAi .u
iœI Ai is the set of elements that belong to every one of the setsAi .
i i this property OeAi for all c
f E o
Ai Azwhatabout Ar forevy real number r
iv Ai EE U Aiie IN
VA 30 A z UA UAou a
Iv Aii e I
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Index sets exampleLet C be the set of Counties in the state of Connecticut (there are 8of these). For each county c œ C , let T (c) be the set of Towns inthat County.
For example, if c is Tolland County, then the elements of T (c) areAndover, Bolton, Columbia, Coventry, Ellington, Hebron, Mansfield,Somers, Sta�ord, Tolland, Union, Vernon, and Willington.
What ist
cœC T (c)?
Te Ic towins in countyCf
UacCTCc elements on any of the
counties
all towns in CT
Tle of
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Index setsLet
R+ = {r : r œ R, r > 0}.
For every real number r œ R+, let
Ar = {(x , y) œ R2 : x2 + y2 < r2}.
op
Ai Ar Fifolfradwsr in plane
Az 5 4
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Index setsWhat is
urœR+ Ar ?
what.Ifang par cx.gl nArfaevyrox2tyLr2 for every re IRt
Inside circle of radiusr 20
r no matter how small r gets
0,0 E AreqAr 0,07 cAr because02 tour
xyl cloore rk Hy oar 2then Cx y Art so not in AreetAn
ftpfr co o
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Index SetsWhat is
trœR+ Ar ? ex yi i
i
i l
Eneypoint Cx y c Ar one a r2 x 4 y 2so evey point'tRin U Ar
r CRtso 112 Yee.fr
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ExampleWhat is
uiœN[0, i + 1]?
A 502 Co 2 ER
Az 0,3Az 0,4 I
A EN Cosi't A EAzEAzEE IN
XEA is cµ
0 its
if Xe Nco itD Heh XE 0,2
A Cositi
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ExampleSuppose that I and J are sets, that J ”= ÿ, and that I ™ J . Is
‹
aœIAa ™
‹
aœJAa?
Explain.
J EIgod
we know
nothingaboutthe
sets Aa
I I 2,3J 1,2
5 10 and TEI
Wehave A Az A 3sets
A Aa A A Az has
aAa A NA
In this case the question is
IS AMALIA E A in Azis
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Left side X XEA and X CAz and Xf 73
Rightside x Xc A and X cAz
so Hellfire A AAzhAz E A A Az
I LT J x XE Z X is even
J EI
Isa zAa
E ane Aa
LHS is A NA an A chAo NA n
x x cAn fr every n e 2JRHS is A yAA 2nA onAzNAYA
x x c An freevy even n C I
Belonging to evy An fu n C I fecesyou to belong to avg Aa for u CZand
n even
General care
if belongs to Aa for evyae IHen since J EI X belongs to Aa for everyones
tafff.EE Aa